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Instability of an inverse problem for the stationary radiative transport near the diffusion limit

In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean free path (the Knudsen number) $0 < \eps \ll 1$. It is well-known that there is a transition of stability from Hölder type to logarithmic type with $\eps\to 0$, the theory of this transition of stability is still an open problem. In this study, we show the transition of stability by establishing the balance of two different regimes depending on the relative sizes of $\eps$ and the perturbation in measurements. When $\eps$ is sufficiently small, we obtain exponential instability, which stands for the diffusive regime, and otherwise we obtain Hölder instability instead, which stands for the transport regime.